Symmetric Matching Pennies Games

audit to obtain the good outcome with a payoff of 1, which is greater than the payoff of –1 for getting caught unprepared. This preferred deviation is represented by the “up” arrow in the lower-left box. Alternatively, if the manager expects no audit (the right side of the table), then the manager prefers not to prepare, which again yields a payoff of 1 (notice the down arrow in the top-right box). Conversely, the auditor prefers to audit when the manager is not prepared, and to skip the audit otherwise.

Given the intuition about being unpredictable, it is not surprising that we typically observe a near-equal split on aggregate decisions for the symmetric matching pennies game in Table 5.1, although there can be considerable variation in the choice proportions from round to round. (In laboratory experiments, the payoffs are scaled up, and a fixed payment is added to eliminate the possibility of losses, but the essential structure of the game is unchanged, as will be seen in Chapter 11.) Despite the intuitive nature of “fifty-fifty” splits for the game in Table 5.1, it is useful to see what behavior is stable in the sense of being a Nash equilibrium. Up to now, we have only considered strategies without random elements, but such strategies will not constitute an equilibrium in this game. First consider the Top/Left box in the table, which corresponds to audit/prepare. This would be a Nash equilibrium if neither player has an incentive to change unilaterally, which is not the case, since the auditor would not want to audit if the manager is going to prepare. In all cells, the player with the lower payoff would prefer to switch unilaterally.

As mentioned in Chapter 1, Nash (1948) proved that an equilibrium always exists (for games in which each person has a finite number of strategies).

Since there is no equilibrium in non-random strategies in the matching pennies game, there must be an equilibrium that involves random play. The earlier discussion of matching pennies already indicated that this equilibrium involves using each strategy half of the time. Think about the “announcement test.” If one person is playing Heads half of the time, then playing Tails will win half of the time; playing Heads will win half of the time, and playing a “fifty-fifty” mix of Heads and Tails will win half of the time. In other words, when one player is playing randomly with probabilities of one half, the other person cannot do any better than using the same probabilities. If each person were to announce that they would use a coin flip to decide which side to play, the other could not do any better than using a coin flip. This is the Nash equilibrium for this game. It is called a “mixed equilibrium” since players use a probabilistic mix of each of their

indifferent, and vice versa. The only way one person will be indifferent is if the other is using equal probabilities of Heads and Tails, which is the equilibrium outcome.

Even though the answer is obvious, it is useful to introduce a graphical device that will help clarify matters in more complicated situations. This graph will show each person’s best response to any given beliefs about the other’s decisions. In Figure 5.1, the thick solid line shows the best response for the row player (manager). The horizontal axis represents what Row expects Column to do. These beliefs can be thought of as a probability of Right, going from 0 on the left to 1 on the right. If Column is expected to choose Left, then Row’s best response is to choose Top, so the best response line starts at the top-left part if the figure, as shown by the thick line. If Column is expected to choose Right, then Row should play bottom, so the best response line ends up on the bottom-right side of the graph. The crossover point is where the Column’s probability is exactly 0.5, since Row does better by playing Top whenever Column is more likely to choose Left.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability of Right

Probability of Top Row's Best Response

Column's Best Response

Figure 5.1. Row’s Best Response to Beliefs about Column’s Decision Column’s Best Response to Beliefs about Row’s Decision

A mathematical derivation of the crossover point (where Row is indifferent and willing to cross over) requires that we find the probability of Right for which Row’s expected payoff is exactly equal for each decision. Let p denote

Row’s beliefs about the probability of Right, so 1– p is the probability of Left.

Recall from the previous chapter that expected payoffs are found by adding up the products of payoffs and associated probabilities. From the top row of Table 5.1, we see that if Row chooses Top, then Row earns 1 with probability 1– p and Row earns –1 with probability p, so the expected payoff is:

Row’s Expected Payoff for Top = 1(1–p) – 1(p) = 1 – 2p.

Similarly, by playing Bottom, Row earns –1 with probability 1– p and 1 with probability p, so the expected payoff is:

Row’s Expected Payoff for Bottom = – (1–p) + p = –1 + 2p.

These expected payoffs are equal when:

1 – 2p = –1 + 2p.

Solving, we see that p = 2/4 = 0.5, which confirms the earlier conclusion that Row is indifferent when Column is choosing each decision with equal probability.

A similar analysis shows that Column is indifferent when Row is using probabilities of one half. Therefore the best response line will “cross over” when Row’s probability of Top is 0.5. To see this graphically, change the interpretation of the axes in Figure 5.1 to let the vertical axis represent Column’s beliefs about what Row will do. And instead of interpreting the horizontal axis as a probability representing Row’s beliefs about Column’s action, now interpret it in terms of Column’s actual best response. With this change, the dashed line that crosses at a height of one half is Column’s best response to beliefs on the vertical axis. If Column thinks Row will play Bottom, then Column wants to play Left, so this line starts in the bottom/left part of the figure. This is because the high payoff of 1 for Column is in the bottom/left part of the payoff table in Table 5.1. There is another payoff of 1 for Column in the top/right part of the table, i.e. when Column thinks Row will play Top, then Right is the best response. For this reason, the dashed best response line in Figure 5.1 ends up in the top/right corner.

In a Nash equilibrium, neither player can do better by deviating, so a Nash equilibrium must be on the best response lines for both players. The only intersection of the solid and dashed lines in Figure 5.1 is at probabilities of 0.5 for

and interpreting the notion of an equilibrium in randomized strategies, not on summarizing all behavioral tendencies in these games.